It turns out that I was using Microsoft's Oracle provider, not Oracle's. Make sure you install Oracle's software and take care to select the right provider when creating a connection. Here's an example connection string for ASP.NET 3.5:

"I would recommend using Oracle Developer Tools (ODT) for VS and ODP.NET since these products are much better integrated with .NET and Oracle.

ODP.NET uses Oracle.DataAccess.Client namespace. System.Data.OracleClient belongs to the MS provider for Oracle, which Mark pointed out. If you drag and drop DB objects using ODT, it will generate ODP.NET code for you (and give you more data source wizard options as well).